Divisor ampleness and quasi-projectivity of group models

By [F1] it suffices for ampleness to exhibit a finite cover of $H$ by affine nonvanishing loci of sections of positive powers of $\mathcal{O}(D)$, or to descend the same statement along a faithfully flat extension. Fibre-density of the complement means that $H\setminus D$ meets every $H^0$-orbit, and after base change to a strict henselization the theorem of the square [F3] gives a linear equivalence $D_g+D_{g^{-1}}\sim 2D$ for suitable translates. The associated sections have nonvanishing loci $gU\cap g^{-1}U$, where $U=H\setminus D$; for a prescribed geometric point, the two conditions on $g$ define dense opens in the geometrically integral $H^0$-fibre. Section values are dense there by [F3], including after extension of its field: an evaluation injection into the product of the fields of section values stays injective after field extension, as can be checked using finitely many linearly independent coefficients. Hence some section $g$ meets both conditions. These loci cover $H$, and quasi-compactness extracts a finite subcover. Each such locus is affine: it is the intersection of two affine opens in the separated scheme over the affine base. The affine-locus definition in [F1] gives ampleness of $\mathcal{O}(D)$ over the strict henselization, and fpqc ampleness descent in [F1] gives it over $R$. This proves (a).

For (b), choose an $R$-dense affine open supplied by the codimension-one neighbourhood lemma and its effective horizontal Cartier boundary $D$; by (a) $\mathcal{O}(D)$ is ample. Choose finitely many affine section opens $X_{s_i}$ covering $H$ and raise the $s_i$ to a common positive degree; each $\mathrm{\Gamma }(X_{s_i},\mathcal{O})$ is a finite-type $R$-algebra with finitely many generators $f_{ij}$, and the section-extension lemma [F2] extends each $f_{ij}{s}_i^n$ to a global section of a sufficiently large common power of $\mathcal{O}(D)$, whose ratios to $s_i^n$ are exactly $f_{ij}$ on $X_{s_i}$.

Including the $s_i^n$ in a finite global section list with no common zero defines a map $H\to {\mathbf{P}}_R^M$ by [F2]; on the projective chart for $s_i^n$ its preimage is $X_{s_i}$ and the coordinate-ring map is surjective because the ratios include all generators $f_{ij}$, so the map is a closed immersion into the union of these projective charts by closed-immersion locality. That union is open in projective space, so the map is a locally closed immersion and $H$ is quasi-projective over $R$; no properness of $H$ is used. This proves (b).